Triangle Lectures in Combinatorics (TLC)
Fifth meeting: February 11, 2012
at Duke University
Speakers:
Alex Fink (NCSU), Matroid Valuations;
Sergey Fomin (Michigan), Cluster structures in rings of SL_3 invariants;
Nets Katz (Indiana University), Erdos Distinct Distances problem in the plane; and
Isabella Novik (University of Washington), Face numbers of centrally symmetric polyopes.
Location: Talks will be in Physics Building Room 128, with coffee breaks in Physics Building Room 101, both at Duke University, Durham, NC
Parking: available in the Bryan Center Parking Garage (PG4) for $5 for the day. This is across the street from the Physics Building.
Schedule:
Friday 4-5pm, NCSU Algebra and Combinatorics Seminar: Leo Mihalcea, Curve Neighborhoods of Schubert Varieties, SAS Hall 4201, NCSU (Raleigh, NC)
Friday 6pm, Seminar Dinner, Details to be determined at the seminar
Saturday, 9:15-10am: coffee and bagels
10-11am, Nets Katz, Erdos Distinct Distance Problem in the Plane
11-11:30am, coffee break
11:30am-12:30pm, Alex Fink, Matroid Valuations
12:30-2:30pm, lunch break
2:30-3:30pm, Isabella Novik, Face Numbers of Centrally Symmetric Polytopes
3:30-4pm, coffee break
4-5pm, Sergey Fomin, Cluster Structures in Rings of SL_3 Invariants
6pm-??, Conference dinner at
Revolution, located at 107 West Main Street (with street parking and nearby parking garage).
To preregister: send email to Patricia Hersh, plhersh@ncsu.edu (there is
no registration fee).
Participant expense reimbursement: There is some limited funding available to reimburse participant hotel and mileage expenses -- if this would help you to be able to attend, please send email to Patricia Hersh, plhersh@ncsu.edu, to apply for funding. Include in this email (1) expected hotel expense for 1-2 nights, (2) total roundtrip mileage (or cost of airfare), (3) a sentence or two of justification, (4) list of other sources of funding available to you. This is primarily intended for participants driving a substantial distance or with compelling reasons it would help their research to be able to stay overnight in the Triangle area.
Hotel suggestions:
Hilton Durham, 919-383-8033, 3800 Hillsborough Rd, Durham, NC (rooms for $130 per night)
Millenium Hotel, 919-383-8495, 2800 Campus Walk Ave, Durham, NC (rooms for $130 per night)
Brookwood Inn, 877-616-2306, 2306 Elba St, Durham, NC (rooms for $100 per night)
Talk titles and abstracts:
Speaker:Alex Fink
Title:Matroid valuations
Abstract:
Many important invariants for matroids, such as the Tutte polynomial,
the Billera-Jia-Reiner quasi-symmetric function, Derksen's universal
invariant, and Speyer's invariant describing the combinatorics of the
Bergman fan, are _valuative_; that is, they are well-behaved under
decompositions of the polytopes.
I will construct the module of matroid valuations, and describe an
explicit combinatorial basis, starting from Brion's theorem; these
results can be extended to polymatroids, and there is a natural Hopf
algebra structure as well.
From a similar starting point, lattice point combinatorics can be used
to associate to every matroid a class in the $K$-theory of the
Grassmannian. This, in particular, provide an algebro-geometric
interpretation of the Tutte polynomial parallel to Speyer's invariant.
Speaker: Sergey Fomin
Title: Cluster structures in rings of SL_3 invariants
Abstract:
The rings of polynomial SL(V)-invariants of configurations of vectors and
linear forms in a k-dimensional complex vector space V have been
explicitly described by Hermann Weyl in the 1930s. Each such ring
conjecturally carries a natural cluster algebra structure (typically, many
of them) whose cluster variables include Weyl's generators. In joint work
with Pavlo Pylyavskyy, we describe and explore these cluster structures in
the case k=3.
Speaker: Nets Katz
Title: Erdos Distinct Distances problem in the plane
Abstract:
In joint work with L. Guth, we show that there is a universal constant
C>0 so that any set of N points in the plane determines at least {N
\over C log N} distinct distances. This settles a longstanding problem
of Erd\"os regarding the best exponent of N that one can obtain in
that estimate.
Speaker: Isabella Novik
Title: Face numbers of centrally symmetric polytopes
Abstract:
How neighborly can a centrally symmetric d-dimensional polytope with n
vertices be? What is the largest number of i-dimensional faces that such a
polytope can have? Which of these polytopes have the largest number of faces? In
this talk I'll report on some recent progress (joint with Alexander Barvinok,
Seung Jin Lee, and Nati Linial) on these questions.
Preregistered participants:
Geir Agnarsson, George Mason University (Fairfax, VA)
Ed Allen, Wake Forest University (Winston-Salem, NC)
Taylor Allison, NCSU (NC)
Cammie Smith Barnes, Sweet Briar College (Sweet Briar, VA)
Christine Berkesch, Duke University (NC)
Sarah Birdsong, UNC Charlotte (Charlotte, NC)
Yue Cai, University of Kentucky (Lexington, KY)
Shaoshi Chen, NCSU (NC)
Ruth Davidson, NCSU (NC)
Alex Fink, NCSU (NC)
Sergey Fomin, Michigan (Ann Arbor, MI)
Jennifer Gamble, NCSU (NC)
Patricia Hersh, NCSU (NC)
Gabor Hetyei, UNC Charlotte (Charlotte, NC)
JT Hird, NCSU (NC)
Johnny Humphries (NC)
John Hutchens, NCSU (NC)
Naihuan Jing, NCSU (NC)
Garrett Johnson, NCSU (NC)
Austin Jones, NCSU (NC)
JiYoon Jung, University of Kentucky (Lexington, KY)
Nets Katz, Indiana University (Bloomington, IN)
Sangwook Kim, George Mason University (Fairfax, VA)
Yonggu Kim, visiting NCSU from Korea (NC)
Patricia Klein, NCSU (NC)
David Lax, UNC Chapel Hill (NC)
Matthew Macauley, Clemson University (Clemson, SC)
Sonja Mapes, Duke University (NC)
Sarah Mason, Wake Forest University (Winston-Salem, NC)
Leo Mihalcea, Virginia Tech (Blacksburg, VA)
Jed Mihalisin, Meredith College (Raleigh, NC)
Ezra Miller, Duke University (NC)
Kailash Misra, NCSU (NC)
Michael Mossinghoff, Davidson College (Davidson, NC)
Liz Munch, Duke University (NC)
Elizabeth Niese, Marshall University (Huntington, WV)
Asamoah Nkwanta, Morgan State University (Baltimore, MD)
Isaballa Novik, Univ. of Washington (Seattle, WA)
Matthew O'Meara, UNC Chapel Hill (NC)
Lindsay Piechnik, High Point University (High Point, NC)
Nathan Reading, NCSU (NC)
Carla Savage, NCSU (NC)
Seth Sullivant, NCSU (NC)
Ryan Vinroot, College of William and Mary (Williamsburg, VA)
Mirko Visontai, U Penn (Philadelphia, PA)
Weikun Wang, NCSU (NC)
Adam Wilkerson, NCSU (NC)
Organizing Committee: Christine Berkesch (Duke), Sonja Mapes (Duke) and Ezra Miller (Duke)